## Applied Math Seminar, Fall 2017

Organized by Wei Zhu

## Time: 3:30 - 4:30 pm, Fridays

Location: 228 Gordon Palmer Hall, Department of Mathematics, University of Alabama

**September 8, 2017****Fei Hu**

Department of Electrical and Computer Engineering, University of Alabama**Title:**Docitive Learning for Wireless Network Spectrum Handoff**Abstract:**Machine learning has been efficiently used in wireless networks to improve their performance. For example, we can use reinforcement learning to make intelligent spectrum allocations in cognitive radio networks (CRN). Docitive learning is a new direction in machine learning field. With the support of U.S. DoD (Department of Defense) , we are able to use docitive learning to make a new joined wireless device learn from an existing device, in order to quickly adapt to the complex wireless conditions. For example, in a CRN, a new device can invoke docitive learning algorithms, to find a good 'teacher' device that can transfer its spectrum handoff strategy to this new node. The Transfer Learning models need to be defined in order to transfer only the most useful protocols to the 'student' device. We propose to use teaching-and-learning models to enhance CRN throughput, that is, the node can accept the teacher's knowledge, but it can also invoke self-learning process if the teacher cannot 'teach' this node anymore due to the different radio environments. We have conducted experiments to verify the efficiency of the docitive learning schemes.**September 15, 2017****Xu Zhang**

Department of Mathematics and Statistics, Mississippi State University**Title:**Immersed Finite Element Methods for Interface Problems Basic idea, Development, Analysis, and Applications**Abstract:**Simulating a multi-scale/multi-physics phenomenon often involves a domain consisting of different materials. This often leads to the so-called interface problems of partial differential equations. Classical finite elements methods can solve interface problems satisfactorily if the mesh is aligned with interfaces; otherwise the convergence could be impaired. Immersed finite element (IFE) methods, on the other hand, allow the interface to be embedded in elements, so that Cartesian meshes can be used for problems with non-trivial interface geometry.

In this talk, we start with an introduction about the basic ideas of IFE methods for the second-order elliptic equation. We will present challenges of conventional IFE methods, and introduce some recent advances in designing more accurate and robust IFE schemes. Mathematical convergence theories and some numerical experiments will be presented. Finally, we will demonstrate how IFE methods can be applied to more complicated interface model problems.**September 22, 2017****Mojdeh Rasoulzadeh**

Department of Mathematics, University of Alabama**Title:**Effective models of flow in highly heterogeneous fractured/vuggy porous media**Abstract:**The presence of vugs and fractures in porous media can significantly affect pressure and flow behavior of a fluid. In this talk, I will present the effective models of flow in a porous medium including multi-scale fractures and several vuggs of various spheroidal shapes. The multiscale fractured porous medium is an infinite set of self-similar double-porosity media. At each scale, the medium consists of a highly permeable network of connected channels and low-permeable blocks. The transition to the macroscale is performed by the means of two-scale homogenization technique.The vuggy medium includes spheroidal fluid filled inclusions. The Stokes flow in the vug is coupled to Darcy’s law in the host matrix using the modified Beavers Joseph boundary condition for the curvy contact between vug and matrix. The pressure field and stream function inside and outside the vug is obtained analytically as a result of solution of harmonic and biharmonic equation inside and outside the vug. The analytic form of equivalent permeability of one single vug is obtained in terms of Legendre polynomials. Then, the effective permeability of a vuggy medium including an arbitrary number of vuggs with different geometries is obtained.**September 29, 2017****Brendan Ames**

Department of Mathematics, University of Alabama**Title:**Semidefinite relaxation of the clustering problem and first-order methods for their solution**Abstract:**I will discuss a novel relaxation approach for the graph clustering problem. Although intractable in worst-case, much recent research has established that clusters can be recovered if the underlying network or data is well-behaved. In particular, I will provide conditions on the underlying graph which guarantee that the solution of this relaxation recovers the desired community structure. Subsequently, I will focus on algorithmic approaches for solving this semidefinite program based on the alternating direction method of multipliers and low-rank factorization, and discuss the challenges associated with applying these methods to large-scale networks.**October 6, 2017****Wei Zhu**

Department of Mathematics, University of Alabama**Title:**New augmented Lagrangian method for a curvature dependent segmentation model**Abstract:**Augmented Lagrangian methods (ALMs) have proved to be successful for the minimization of curvature dependent functionals in image processing. However, those ALM based algorithms often suffer from choosing appropriate penalty parameters in the numerical implementation. In this talk, we will discuss our recent work on the development of ALM based algorithms for a curvature based segmentation model by introducing fewer Lagrange multipliers. Besides significantly reducing the effort of selecting suitable penalty parameters, the new algorithms help capture curvature more faithfully than those existing ones. Numerical experiments will also be presented to valid the effectiveness of the proposed algorithms.**October 11, 2017, 3:30pm - 4:30pm, Wednesday, GP 346.****(Colloquium of Mathematics Department)****Zhimin Zhang**

Department of Mathematics, Wayne State University**Title:**Polynomial Preserving Recovery for Gradient and Hessian**Abstract:**Post-processing techniques are important in scientific and engineering computation. One of such technique, Superconvergent Patch Recovery (SPR) proposed by Zienkiewicz-Zhu in 1992, has been widely used in finite element commercial software packages such as Abaqus, ANSYS, Diffpack, etc.; another one, Polynomial Preserving Recovery (PPR) has been adopted by COMSOL Multiphysics since 2008. In this talk, I will give a survey for the PPR method and discuss its resent development to obtain the Hessian matrix (second derivatives) from the computed data.**October 17, 2017**,**3:00pm - 4:00pm, Tuesday, GP 302.****(Colloquium of Mathematics Department)****Ricardo Cortez**

Department of Mathematics, Tulane University**Title:**Mathematical and Computational Modeling of Microorganism

Swimming Motions**Abstract:**Microscopic swimmers like bacteria and spermatozoa live in highly viscous environments. Their locomotion and the fluid flows they generate around them have been actively investigated for the last 60 years motivated by questions about effective locomotion strategies, the organism’s interaction with the surrounding environment, patterns of collective motion, propulsion, and more. These issues are typically addressed through a combination of theory, experiments, mathematical modeling and simulation. I will describe mathematical ideas that led to the “method of regularized Stokeslets,” a computational technique based on fundamental solutions of PDEs designed for simulating viscous flows generated by external forcing. I will show several applications, including work done with undergraduate students, that sheds light on these biological systems and challenges ahead.**October 20, 2017****Aijun Song**

Department of Electrical & Computer Engineering, University of Alabama**Title:**Time reversal acoustic communication in the ocean**Abstract:**The global marine ecosystem is undergoing significant changes due to human activities and natural processes. These changes call for enhanced capabilities to sample and communicate in the oceans. With this background, underwater acoustic communication has attracted much attention across multiple disciplines, as this key subsea technology allows real-time access to oceanographic measurements and supports navigation of underwater vehicles. Through the years, I devote efforts to develop a class of reliable communication algorithms, known as time reversal receivers. In this talk, communication challenges in the ocean are discussed. Time reversal receivers and their performance in the ocean are presented. I will also present our on-going effort to achieve full-duplex underwater acoustic communications.**October 27, 2017****Lin Mu**

Computer Science and Mathematics Division, Oak Ridge National Laboratory**Title:**A Priori and a posteriori error estimate for weak Galerkin finite element method on polygonal meshes**Abstract:**Polygonal mesh has advantages including lower DOFs requirement for the same level of accuracy and more flexibility in generating mesh, and better mesh quality over standard discretization with quad mesh or triangular mesh. Also the hanging nodes are handled naturally, which will introduce more simple refinement strategies when applied into the adaptive finite element methods. Furthermore, because the relax of continuity requirement, the numerical schemes for the PDEs solver can be easily extended to high-order schemes by utilizing polynomials with higher degrees. In this talk, we will discuss about the novel weak Galerkin finite element methods to discrete the PDEs and analyze the a priori and a posteriori error estimate accordingly. The key of this new method is to replace the derivatives by their weak discrete derivatives. The new method is a discontinuous finite element approach, which is parameter free, symmetric, and absolutely stable. The fully computable a posteriori error estimate gives an reliable estimate for the interested error. We will also show the robustness of our approach theoretically and numerically.**November 10, 2017****(Colloquium of Mathematics Department)****Xiaoming Huo**

Stewart School of Industrial & Systems Engineering, Georgia Institute of Technology**Title:**Statistically and Numerically Efficient Independence Test**Abstract:**The big data is a well-known phenomenon in the modern world. The emerging discipline of data science has inspired a lot of discussion and debate in the scientific research communities, including the mathematical and statistical science community. Contributing to this discussion, in the first part of this talk, I will present a discussion as well as a selective survey on the landscape of data science, as it is forming its foundation. On the second part of this talk, I will present one of my own research, which addresses a particular issue in the enormous spectrum of data science. More specifically, we study how to generate a statistical inference procedure that is both computational efficient and having theoretical guarantee on its statistical performance. We present numerical comparisons with contemporary approaches to demonstrate its advantages.